Equidistribution for parameter-dependent rational dynamical systems

Determine conditions on a one-parameter family of rational maps F_λ on projective space, a rational parameter map, and a homogeneous polynomial Σ under which the normalized measures supported on the solutions of (Σ∘F_λ^n∘)(λ)=0 equidistribute toward a probability measure.

Background

The graph-theoretic equidistribution problem is reformulated abstractly for parameter-dependent rational dynamics. This extends classical preimage-equidistribution results for a single rational map and their higher-dimensional analogues, but the paper notes that the corresponding general rational setting remains far from fully understood.

References

Under which conditions on F_λ, , and Σ, do the probability measures νn, given by the normalized sums of the Dirac masses at the solutions of the equation (Σ ∘ Fλn ∘ )(λ) = 0, equidistribute toward a probability measure μ?

— Zeros of the independence polynomial on recursive sequences of graphs  (2609.35102 - Hlushchanka et al., 28 Sep 2026) in Section 1, subsection “Open questions,” subsubsection “Equidistribution of zeros”